Assurance and annuity functions involving two lives: formula sheet

Key formulas

Joint life survival (independent lives)

tpxy = tpx × tpy

Both must survive t years. Needs independence.

Joint life death probability

tqxy = 1 − tpxy = 1 − tpx × tpy

At least one dies within t years.

Last survivor survival

tpx̄ȳ = tpx + tpy − tpxy

At least one survives t years. Valid for any two lives, since it is the addition rule for events.

Last survivor death probability (independent lives)

tqx̄ȳ = tqx × tqy

Both die within t years. Needs independence.

Complement form

tpx̄ȳ = 1 − tqx × tqy

Quick route for independent lives.

Status lifetimes

T_xy = min(T_x, T_y); T_x̄ȳ = max(T_x, T_y)

Gives T_xy + T_x̄ȳ = T_x + T_y.

Exactly one survives (independent lives)

tpx × tqy + tqx × tpy

Equals tpx̄ȳ − tpxy.

Time to first and last death

T_xy = min(T_x, T_y); T_x̄ȳ = max(T_x, T_y)

Also T_xy + T_x̄ȳ = T_x + T_y. The curtate versions are K_xy = min(K_x, K_y) and K_x̄ȳ = max(K_x, K_y).

Joint life survival probability

t p_xy = t p_x × t p_y

Needs independence. Then t q_xy = 1 − t p_x × t p_y.

Last survivor survival probability

t p_x̄ȳ = t p_x + t p_y − t p_xy

Holds for any joint distribution of (T_x, T_y). Under independence, t q_x̄ȳ = t q_x × t q_y.

Force of mortality of the joint status

μ_xy(t) = μ_x+t + μ_y+t

Under independence. Density of T_xy: f(t) = t p_xy × (μ_x+t + μ_y+t).

Density of the time to last death

f(t) = t p_x μ_x+t + t p_y μ_y+t − t p_xy (μ_x+t + μ_y+t)

Differentiate −t p_x̄ȳ with respect to t. Under independence, this equals the derivative of t q_x × t q_y.

Complete expectation of life

e°_xy = ∫₀^∞ t p_xy dt; e°_x̄ȳ = e°_x + e°_y − e°_xy

e°_x̄ȳ uses t p_x̄ȳ in the integral. The relation follows because t p_x̄ȳ = t p_x + t p_y − t p_xy.

Curtate expectation of life

e_xy = Σ (k = 1 to ∞) k p_xy; e_x̄ȳ = e_x + e_y − e_xy

e_x̄ȳ uses k p_x̄ȳ in the sum. Always start the sum at k = 1.

Probability of death in a given year

P(K_xy = k) = k p_xy − (k+1) p_xy

The same form works for the last survivor with k p_x̄ȳ. K is the curtate future lifetime.

Approximation linking curtate and complete

e°_xy ≈ e_xy + ½

Valid only under the assumption of uniform distribution of deaths within each year. It is an approximation, not an identity.

Joint life status failure time

T_xy = min(T_x, T_y)

The status (xy) fails on the first death.

Last survivor status failure time

T_x̄ȳ = max(T_x, T_y)

The status (x̄ȳ) fails on the second death.

Whole life assurance on a status (discrete)

A_xy = E[v^(K_xy + 1)] and A_x̄ȳ = E[v^(K_x̄ȳ + 1)]

K is the curtate future lifetime of the status. Payment is at the end of the year of failure.

Whole life assurance on a status (continuous)

Ā_xy = E[v^T_xy] and Ā_x̄ȳ = E[v^T_x̄ȳ]

Payment at the moment the status fails.

Key relationship

A_xy + A_x̄ȳ = A_x + A_y

Also Ā_xy + Ā_x̄ȳ = Ā_x + Ā_y. Holds for any dependence between the lives, using the same basis for all four.

Last survivor from the others

A_x̄ȳ = A_x + A_y − A_xy

Use this to avoid computing second-death probabilities directly.

Survival probabilities

tp_xy = tp_x × tp_y (independent lives); tp_x̄ȳ = tp_x + tp_y − tp_xy

The first needs independence. The second always holds.

Term assurance on a status

Joint life: A¹_xy:n = Σ (k = 0 to n−1) v^(k+1) × kp_xy × q_(x+k:y+k). Last survivor: A¹_x̄ȳ:n = Σ (k = 0 to n−1) v^(k+1) × (kp_x̄ȳ − (k+1)p_x̄ȳ)

The first formula is for the joint life status only. Here q_(x+k:y+k) = 1 − p_(x+k:y+k) is the probability the joint status fails in the year, so the first-death probability in year k+1 is kp_xy × q_(x+k:y+k). For the last survivor status, do not use this expression. Use the status's own survival probabilities: the probability it fails in year k+1 is kp_x̄ȳ − (k+1)p_x̄ȳ.

Recursion for joint life assurance

A_xy = v × q_xy + v × p_xy × A_(x+1:y+1)

One-year recursion for the joint life status, with q_xy = 1 − p_xy.

Joint life annuity-due

ä_xy = Σ (t = 0 to ∞) v^t · t p_xy

Payments at the start of each year while both lives are alive. With independent lives, t p_xy = t p_x · t p_y.

Last survivor survival probability

t p_x̄ȳ = t p_x + t p_y − t p_xy

Probability that at least one life survives t years.

Last survivor annuity-due

ä_x̄ȳ = ä_x + ä_y − ä_xy

Holds for any joint distribution of the two lives.

Annuity-immediate relation

a_xy = ä_xy − 1, and a_x̄ȳ = ä_x̄ȳ − 1

Applies for whole life annuities. The first payment is at time 0 for the due form.

Continuous joint life annuity

ā_xy = ∫ (0 to ∞) v^t · t p_xy dt

Payable continuously. The identity ā_x̄ȳ = ā_x + ā_y − ā_xy also holds.

Equivalence principle premium

P · (EPV of 1 per year premium annuity) = EPV of benefits

Use the premium annuity status specified by the contract.

Reversionary annuity (annual, in arrears)

a_x|y = a_y − a_xy

Paid to (y) while y is alive and x is dead. Joint-life annuity a_xy is subtracted from the single-life annuity on (y).

Reversionary annuity (annuity-due and continuous forms)

ä_x|y = ä_y − ä_xy and ā_x|y = ā_y − ā_xy

Use the same type of annuity on both sides. Do not mix due and immediate values.

Joint life and last survivor annuities

a_xy = value while both alive; a_(x̄ȳ) = a_x + a_y − a_xy

The last survivor annuity pays while at least one life is alive. It is not the same as a reversionary annuity.

Decomposition of the last survivor annuity

a_(x̄ȳ) = a_xy + a_x|y + a_y|x

Both alive, only y alive, only x alive. Useful as a check on your numbers.

Contingent assurance, death of x first (continuous)

A^1_xy = ∫₀^∞ v^t · t p_xy · μ_(x+t) dt

Pays 1 at the moment of death of (x) if (y) is still alive. Under independence, t p_xy = t p_x × t p_y.

Contingent assurance, death of y second (continuous)

A^2_xy = ∫₀^∞ v^t · t p_y · μ_(y+t) · (1 − t p_x) dt

Pays 1 on the death of (y) if (x) has died earlier. This form assumes independent lives.

Order-of-death identities

A^1_xy + A^1_yx = A_xy and A_y = A^1_yx + A^2_xy, so A^2_xy = A_y − A^1_yx

The first identity splits the first-death assurance by who dies first. The second splits the assurance on (y) by whether y dies first or second.

Quick revision

• Joint life status xy fails at the first death. Last survivor status x̄ȳ fails at the second death.
• Under independence, tpxy = tpx × tpy.
• Under independence, tp x̄ȳ = tpx + tpy − tpx × tpy.
• Time to first death plus time to last death equals Tx + Ty, so last survivor functions equal the sum of single-life functions minus the joint life function.
• So ax̄ȳ = ax + ay − axy, and the same pattern holds for assurances.
• Annuity-due on a status: ä = Σ v^t × tp(status) for t = 0, 1, 2, ...
• A reversionary annuity paid to (y) after the death of (x) has value ay − axy.
• Contingent assurances pay only if a named life dies first. Check the order of death carefully.
• State independence explicitly unless the question gives dependence.
• Write each status before writing a formula, and check the payment timing is right.
• Check answers are sensible: last survivor annuity ≥ either single life annuity ≥ joint life annuity.

Common mistakes

• Treating the last survivor status as failing at the first death. Fix: Joint life (xy) fails at the first death. Last survivor (x̄ȳ) fails at the second death.
• Using tpx̄ȳ = tpx × tpy. Fix: The product gives the joint life only. For last survivor use tpx + tpy − tpxy.
• Using t p_xy = t p_x + t p_y or taking the average of the two lifetimes. Fix: Joint life means both alive, so multiply (under independence). The addition-and-subtract form belongs to the last survivor only.
• Writing t q_x̄ȳ = t q_x + t q_y. Fix: A last survivor status has failed only when both lives are dead, so multiply the death probabilities: t q_x̄ȳ = t q_x × t q_y (independence).
• Swapping which status gives first death and which gives second death. Fix: Write: no bar (xy) = both alive = pays on first death. Bar (x̄ȳ) = either alive = pays on second death. Then check A_xy ≥ A_x̄ȳ.
• Writing A_x̄ȳ = A_x × A_y or A_xy = A_x × A_y. Fix: Assurance values are expectations of v^T, not probabilities. The product of v^Tx and v^Ty is not v^min. Use the sum identity instead.
• Using ä_x + ä_y as the last survivor value Fix: Always subtract ä_xy: ä_x̄ȳ = ä_x + ä_y − ä_xy.
• Writing t p_xy = t p_x + t p_y − something Fix: Joint life is an intersection: product when independent. Last survivor is a union: add and subtract the intersection.
• Treating a_x|y as paying to x Fix: Read a_x|y as 'to (y) after (x) dies'. The payee is the life after the bar. The life before the bar must die first.
• Calling a reversionary annuity a last survivor annuity Fix: A last survivor annuity pays while at least one life is alive, including when both are alive. A reversionary annuity pays only after the named life dies, to the other life. Nothing is paid while both are alive.

Exam tips

• Write the status in words first. Marks are often lost by picking the wrong status, not by arithmetic.
• State 'assuming independence' in written answers. Examiners give credit for stating assumptions.
• Use the identity tpxy + tpx̄ȳ = tpx + tpy as a quick check, and to find one status when the other is given.
• In MCQs, use the complement route 1 − tqx × tqy for last survivor survival. It is the fastest and has fewest errors.
• Show age, term and notation in every line of working, for example 10p60, so the marker can follow it.
• Write the independence assumption at the top of every answer. It earns a mark and shows you know when the product rule applies.
• In MCQs, check the status first. Many wrong options are the right formula for the other status.
• Use the sanity bounds: e_xy ≤ min(e_x, e_y) and e_x̄ȳ ≥ max(e_x, e_y). They catch arithmetic slips fast.